Why the Geomantic Judge is Always Even

[sdi]

You cast sixteen random numbers (each 1 or 2) to generate the mothers of a geomantic chart. Let's call them A, B, C, D, etc. To generate the daughters, you simply rearrange these. To generate the nieces and the court, you add these together (modulo 2) in a pairwise fashion:

D H L P	C G K O	B F J N	A E I M	M N O P	I J K L	E F G H	A B C D
C+D G+H K+L O+P		A+B E+F I+J M+N		I+M J+N K+O L+P		A+E B+F C+G D+H
A+B+C+D E+F+G+H I+J+K+L M+N+O+P				A+E+I+M B+F+J+N C+G+K+O D+H+L+P
A+A+B+C+D+E+I+M B+E+F+F+G+H+J+N C+G+I+J+K+K+L+O D+H+L+M+N+O+P+P

What happens if we add each of the lines of the judge together? We would end up with A+A+B+B+C+C+D+D+...: that is, each number appears exactly twice. (This is because one copy comes from the mothers and the other comes from the daughters.)

Now, consider what it means to add X+X in geomancy. An active line has one dot, so 1+1=2. A latent line has two dots, so 2+2=4=2. So if you ever add a number to itself, you always get 2.

So A+A+B+B+C+C+D+D+...=2+2+2+2+...=2, regardless of what A, B, C, D, etc. are. Therefore, your judge will always have an even number of dots. This means it will always be one of Populus, Fortuna Minor, Amissio, Conjunctio, Carcer, Acquisitio, Fortuna Major, or Via. ∎